Jacobi method

(Difference between revisions)
 Revision as of 07:54, 8 April 2006 (view source)Jasond (Talk | contribs) (Added an example calculation here, too)← Older edit Revision as of 10:08, 12 May 2007 (view source)Jola (Talk | contribs) Newer edit → Line 1: Line 1: - == Introduction == + The Jacobi method is an algorithm in linear algebra for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after German mathematician [http://en.wikipedia.org/wiki/Carl_Gustav_Jakob_Jacobi Carl Gustav Jakob Jacobi]. We seek the solution to set of linear equations:
We seek the solution to set of linear equations:
Line 16: Line 16: [/itex] [/itex] - Note that the computation of $\phi^{(k+1)}_i$ requires each element in $\phi^{(k)}$ except itself.  Then, unlike in the [[Gauss-Seidel method]], we can't overwrite $\phi^{(k)}_i$ with $\phi^{(k+1)}_i$, as that value will be needed by the rest of the computation.  This is the most meaningful difference between the Jacobi and Gauss-Seidel methods.  The minimum ammount of storage is two vectors of size $n$, and explicit copying will need to take place. + Note that the computation of $\phi^{(k+1)}_i$ requires each element in $\phi^{(k)}$ except itself.  Then, unlike in the [[Gauss-Seidel method]], we can't overwrite $\phi^{(k)}_i$ with $\phi^{(k+1)}_i$, as that value will be needed by the rest of the computation.  This is the most meaningful difference between the Jacobi and Gauss-Seidel methods.  The minimum amount of storage is two vectors of size $n$, and explicit copying will need to take place. == Algorithm == == Algorithm == - Chose an initial guess $\phi^{0}$ to the solution
+ Choose an initial guess $\phi^{0}$ to the solution
- :    for k := 1 step 1 untill convergence do
+ :    for k := 1 step 1 until convergence do
::  for i := 1 step until n do
::  for i := 1 step until n do
:::  $\sigma = 0$
:::  $\sigma = 0$
Line 34: Line 34: ==Convergence== ==Convergence== - It is proven that if the absolute value of the diagonal term is always greater than the sum of the absolute values of other term in the row: + The method will always converge if the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms: + :$\left | a_{ii} \right | > \sum_{i \ne j} {\left | a_{ij} \right |}$ :$\left | a_{ii} \right | > \sum_{i \ne j} {\left | a_{ij} \right |}$ - then the method always converge. - Usually, but not always, the method converges even if this condition is not satisfied, but the diagonal terms in the matrix are greater by the absolute values than the other terms. + The Jacobi method sometimes converges even if this condition is not satisfied. It is necessary, however, that the diagonal terms in the matrix are greater (in magnitude) than the other terms. == Example Calculation == == Example Calculation == Line 152: Line 152: - ==External link== + ==External links== - *[http://en.wikipedia.org/wiki/Jacobi_method Wikipedia article] + *[http://www.math-linux.com/spip.php?article49 Jacobi method from www.math-linux.com] + *[http://mathworld.wolfram.com/JacobiMethod.html Jacobi Method at Math World] + *[http://en.wikipedia.org/wiki/Jacobi_method Jacobi method at Wikipedia]

Revision as of 10:08, 12 May 2007

The Jacobi method is an algorithm in linear algebra for determining the solutions of a system of linear equations with largest absolute values in each row and column dominated by the diagonal element. Each diagonal element is solved for, and an approximate value plugged in. The process is then iterated until it converges. This algorithm is a stripped-down version of the Jacobi transformation method of matrix diagonalization. The method is named after German mathematician Carl Gustav Jakob Jacobi.

We seek the solution to set of linear equations:

$A \phi = b$

In matrix terms, the definition of the Jacobi method can be expressed as :

$\phi^{(k+1)} = D^{ - 1} \left[\left( {L + U} \right)\phi^{(k)} + b\right]$

where $D$, $L$, and $U$ represent the diagonal, lower triangular, and upper triangular parts of the coefficient matrix $A$ and $k$ is the iteration count. This matrix expression is mainly of academic interest, and is not used to program the method. Rather, an element-based approach is used:

$\phi^{(k+1)}_i = \frac{1}{a_{ii}} \left(b_i -\sum_{j\ne i}a_{ij}\phi^{(k)}_j\right),\, i=1,2,\ldots,n.$

Note that the computation of $\phi^{(k+1)}_i$ requires each element in $\phi^{(k)}$ except itself. Then, unlike in the Gauss-Seidel method, we can't overwrite $\phi^{(k)}_i$ with $\phi^{(k+1)}_i$, as that value will be needed by the rest of the computation. This is the most meaningful difference between the Jacobi and Gauss-Seidel methods. The minimum amount of storage is two vectors of size $n$, and explicit copying will need to take place.

Algorithm

Choose an initial guess $\phi^{0}$ to the solution

for k := 1 step 1 until convergence do
for i := 1 step until n do
$\sigma = 0$
for j := 1 step until n do
if j != i then
$\sigma = \sigma + a_{ij} \phi_j^{(k-1)}$
end if
end (j-loop)
$\phi_i^{(k)} = {{\left( {b_i - \sigma } \right)} \over {a_{ii} }}$
end (i-loop)
check if convergence is reached
end (k-loop)

Convergence

The method will always converge if the matrix A is strictly or irreducibly diagonally dominant. Strict row diagonal dominance means that for each row, the absolute value of the diagonal term is greater than the sum of absolute values of other terms:

$\left | a_{ii} \right | > \sum_{i \ne j} {\left | a_{ij} \right |}$

The Jacobi method sometimes converges even if this condition is not satisfied. It is necessary, however, that the diagonal terms in the matrix are greater (in magnitude) than the other terms.

Example Calculation

As with Gauss-Seidel, Jacobi iteration lends itself to situations in which we need not explicitly represent the matrix. Consider the simple heat equation problem

$\nabla^2 T(x) = 0,\ x\in [0,1]$

subject to the boundary conditions $T(0)=0$ and $T(1)=1$. The exact solution to this problem is $T(x)=x$. The standard second-order finite difference discretization is

$T_{i-1}-2T_i+T_{i+1} = 0,$

where $T_i$ is the (discrete) solution available at uniformly spaced nodes (see the Gauss-Seidel example for the matrix form). For any given $T_i$ for $1 < i < n$, we can write

$T_i = \frac{1}{2}(T_{i-1}+T_{i+1}).$

Then, stepping through the solution vector from $i=2$ to $i=n-1$, we can compute the next iterate from the two surrounding values. For a proper Jacobi iteration, we'll need to use values from the previous iteration on the right-hand side:

$T_i^{k+1} = \frac{1}{2}(T_{i-1}^{k}+T_{i+1}^k).$

The following table gives the results of 10 iterations with 5 nodes (3 interior and 2 boundary) as well as $L_2$ norm error.

Jacobi Solution
Iteration $T_1$ $T_2$ $T_3$ $T_4$ $T_5$ $L_2$ error
0 0.0000E+00 0.0000E+00 0.0000E+00 0.0000E+00 1.0000E+00 1.0000E+00
1 0.0000E+00 0.0000E+00 0.0000E+00 5.0000E-01 1.0000E+00 6.1237E-01
2 0.0000E+00 0.0000E+00 2.5000E-01 5.0000E-01 1.0000E+00 4.3301E-01
3 0.0000E+00 1.2500E-01 2.5000E-01 6.2500E-01 1.0000E+00 3.0619E-01
4 0.0000E+00 1.2500E-01 3.7500E-01 6.2500E-01 1.0000E+00 2.1651E-01
5 0.0000E+00 1.8750E-01 3.7500E-01 6.8750E-01 1.0000E+00 1.5309E-01
6 0.0000E+00 1.8750E-01 4.3750E-01 6.8750E-01 1.0000E+00 1.0825E-01
7 0.0000E+00 2.1875E-01 4.3750E-01 7.1875E-01 1.0000E+00 7.6547E-02
8 0.0000E+00 2.1875E-01 4.6875E-01 7.1875E-01 1.0000E+00 5.4127E-02
9 0.0000E+00 2.3438E-01 4.6875E-01 7.3438E-01 1.0000E+00 3.8273E-02
10 0.0000E+00 2.3438E-01 4.8438E-01 7.3438E-01 1.0000E+00 2.7063E-02